Stochastic Diversity Stabilization in Nonlinear Competitive Dynamics: A Statistical Mechanics Approach to Fluctuation Suppression and Finite-Size Scaling
Stochastic forcing can prevent diversity collapse in competitive dynamical systems, yet the quantitative conditions and the dynamical mechanism responsible remain poorly characterized. We address both questions in a class of nonlinear stochastic replicator dynamics on the N-dimensional probability simplex, parameterized by a nonlinearity exponent gamma and additive Gaussian noise of intensity sigma. Under the baseline interaction, the minimum noise required to sustain diversity decays as a power law, sigma_c ~ N^(-alpha); replacing the baseline interaction with a superlinear variant shifts alpha from about 1.1 to about 2.0, in agreement with a linear-stability prediction alpha = (gamma+2)/2, identifying alpha as a model-specific quantity rather than a universal index. Scanning the (gamma, sigma) plane reveals a smooth dynamical crossover: the fluctuation susceptibility chi_H = N Var(H) peaks at gamma ~ 0.7 (chi_H ~ 0.6) and is suppressed at high gamma, with no divergence near gamma ~ 1.4; matched finite-size analysis finds no systematic divergence of the peak with N, ruling out a genuine second-order phase transition. The entropy relaxation time tau_relax(gamma) remains approximately constant (about 2.9 time units) across the full range of gamma, even as chi_H decreases sixfold; this decoupling between relaxation timescale and fluctuation amplitude implies that suppression arises from convergence of diverse pathways to similar endpoints rather than from accelerated relaxation along a single channel. These findings identify nonlinear fluctuation suppression via pathway multiplicity -- distinct from both noise-induced transitions and equilibrium criticality -- as the operative stabilization pattern in this class of nonequilibrium stochastic systems.